Two Weight Inequality for the Hilbert Transform: A Real Variable Characterization, I

TL;DR

This paper provides a real-variable characterization of the two weight inequality for the Hilbert transform, showing it holds if and only if certain L^2 and weak-L^2 inequalities are satisfied, with implications for operator theory.

Contribution

It offers a new characterization of the two weight inequality for the Hilbert transform using two-weight Poisson and testing inequalities, simplifying previous complex conditions.

Findings

Two weight inequality holds iff two L^2 to weak-L^2 inequalities hold.

Characterization involves two-weight Poisson inequality and testing inequalities.

Results simplify understanding of the two weight inequality for the Hilbert transform.

Abstract

The two weight inequality for the Hilbert transform arises in the settings of analytic function spaces, operator theory, and spectral theory, and what would be most useful is a characterization in the simplest real-variable terms. We show that the $L^2$ to $L^2$ inequality holds if and only if two L^2 to weak-L^2 inequalities hold. This is a corollary to a characterization in terms of a two-weight Poisson inequality, and a pair of testing inequalities on bounded functions.